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A company operates a solar installation in the desert in WesternAustralia. It is reviewing its operating practices with a view tomaking them more efficient

. a) The solar installation generates electric power fromsunlight and incurs operating costs for cleaning the solar modules(sometimes called solar panels) and replacing solar modules thathave failed. The annual revenue from the electric power is variabledue to variable cloudiness and solar module failure and has a meanof $2.78m and a standard deviation of $0.32m. The annual operatingcosts have a mean of $0.51m and a standard deviation of $0.12m.Calculate the mean and standard deviation of the annual profit =annual revenue – annual operating costs.

b) Expected revenue varies systematically from one month toanother, being higher in the summer when there is more sunshine.Monthly operating costs follow the same probability modelregardless of the month (same mean and standard deviation apply toall months). Calculate, if possible, the mean and standarddeviation of (i) monthly operating costs (ii) monthly profits. If acalculation is not possible, give the reason.

c) The solar installation is located in the desert 100 km fromthe nearest office of the company that operates it and the companysends a maintenance crew out quarterly (once every 3 months) toclean dust and sand off the solar modules and check for mechanicalor electrical problems. Each solar module is also monitoredelectronically over the Internet so that the operating company isalerted immediately when a solar module fails. On average 1.3modules fail per month and the maintenance crew replaces any failedmodules on their quarterly visits. Module failures are independentof each other and occur at random. The loss of a few solar modulesdoes not impact revenue enough to justify the cost of sending themaintenance crew before the next quarterly visit. However theoperating company decides that if more than 7 modules have failedthey should send the maintenance crew out immediately to replacethe failed modules. What is the probability of the maintenance crewhaving to go to the solar installation before the end of theregular 3-month period?

d) If 8 modules fail, the maintenance crew loads 9 replacementmodules into their truck in case one is smashed during the 100 kmdrive, much of which is over uneven dirt tracks through the desert.Past experience shows that the probability of any individual modulebeing smashed on this journey is 0.043. The operations managerwants the probability that the crew arrives with less than 8working modules to be < 0.05. How many replacement modulesshould the maintenance crew load into their truck so as to achievethis objective? Answer this question, stating your assumptionsclearly, and comment on whether the assumptions are likely to betrue.

e) The solar modules are covered by a 25-year warranty whichcovers the cost of the replacement module itself but not the costof driving 100 km and installing it. The operating company plans onvisiting the site only once every 3 months and is thereforeconsidering purchasing “business continuity insurance” which wouldcover the loss of revenue from failed solar modules for an annualpremium of $5000. In order to decide whether it is worth payingthis premium the company needs to calculate its expected revenueloss from failed modules. The average loss of revenue from onefailed module is $200 per month. If one module fails during a3-month period, we assume it fails in the middle of that period sothat it has failed for a total of 1.5 months and the loss ofrevenue is 1.5*200 = $300. We make similar assumptions if 2,3,4, …modules fail during the 3 month period. Considering theprobabilities of 0,1,2, …,10 modules failing during a 3-monthperiod, what is the expected revenue loss during a 3-month period?Based on this expected loss, should the company purchase businesscontinuity insurance?